In crystallography, a fractional coordinate system (crystal coordinate system) is a coordinate system in which basis vectors used to describe the space are the lattice vectors of a crystal (periodic) pattern. The selection of an origin and a basis define a unit cell, a parallelotope (i.e., generalization of a parallelogram (2D) or parallelepiped.

Most cases in crystallography involve two- or three-dimensional space.

Crystal Structure

A crystal structure is defined as the spatial distribution of the atoms within a crystal, usually modeled by the idea of an infinite crystal pattern. An infinite crystal pattern refers to the infinite 3D periodic array which corresponds to a crystal, in which the lengths of the periodicities of the array may not be made arbitrarily small. The geometrical shift which takes a crystal structure coincident with itself is termed a symmetry translation (translation) of the crystal structure. The vector which is related to this shift is called a translation vector. Since a crystal pattern is periodic, all integer linear combinations of translation vectors are also themselves translation vectors,

Lattice

The vector lattice is defined as the infinite set consisting of all of the translation vectors of a crystal pattern. Each of the vectors in the vector lattice are called lattice vectors. From the vector lattice it is possible to construct a point lattice. This is done by selecting an origin can be chosen and paired with the lattice vectors of the vector lattice. The points or particles that are made coincident with one another through a translation are called translation equivalent.

Coordinate systems

General coordinate systems

Usually when describing a space geometrically, a coordinate system is used which consists of a choice of origin and a basis-tuple is then,

respectively.

Cartesian coordinate system

A widely used coordinate system is the Cartesian coordinate system, which consists of orthonormal basis vectors. This means that,

However, when describing objects with crystalline or periodic structure a Cartesian coordinate system is often not the most useful as it does not often reflect the symmetry of the lattice in the simplest manner.

Fractional (crystal) coordinate system

In crystallography, a fractional coordinate system is used in order to better reflect the symmetry of the underlying lattice of a crystal pattern (or any other periodic pattern in space). In a fractional coordinate system the basis vectors of the coordinate system are chosen to be lattice vectors and the basis is then termed a crystallographic basis (or lattice basis).

can be represented as,

There are an infinite number of lattice bases for a crystal pattern. However, these can be chosen in such a way that the simplest description of the pattern can be obtained. These bases are used in the International Tables of Crystallography Volume A and are termed conventional bases is called primitive can be expressed as,

However, the conventional basis for a crystal pattern is not always chosen to be primitive. Instead, it is chosen so the number of orthogonal basis vectors is maximized. This results in some of the coefficients of the equations above being fractional. A lattice in which the conventional basis is primitive is called a primitive lattice, while a lattice with a non-primitive conventional basis is called a centered lattice.

The choice of an origin and a basis implies the choice of a unit cell.

Furthermore, points outside of the unit cell can be transformed inside of the unit cell through standardization are called the lattice parameters (lattice constants) of the lattice. In two- and three-dimensions, these correspond to the lengths and angles between the edges of the unit cell.

in terms of the lattice basis vectors is defined as,

Calculations involving the unit cell

General transformations between fractional and Cartesian coordinates

Three Dimensions

The relationship between fractional and Cartesian coordinates. Letting , , , , , and :

Similarly, the Cartesian coordinates:

Transformations using the cell tensor

Another common method of converting between fractional and Cartesian coordinates involves the use of a cell tensor which contains each of the basis vectors of the space expressed in Cartesian coordinates.

Two Dimensions

Cell tensor

In Cartesian coordinates cell tensor:

The area of the unit cell, is given by the determinant of the cell matrix:

For the special case of a square or rectangular unit cell, the matrix is diagonal, and we have that:

Relationship between fractional and Cartesian coordinates

The relationship between fractional and Cartesian coordinates:

Similarly, the Cartesian coordinates:

Three Dimensions

Cell tensor

In Cartesian coordinates cell tensor:

The volume of the unit cell, is given by the determinant of the cell tensor:

For the special case of a cubic, tetragonal, or orthorhombic cell, the matrix is diagonal, and we have that:

Relationship between fractional and Cartesian coordinates

The relationship between fractional and Cartesian coordinates:

Similarly, the Cartesian coordinates:

Arbitrary number of dimensions

Cell tensor

In Cartesian coordinates cell tensor:

The hypervolume of the unit cell, is given by the determinant of the cell tensor:

Relationship between fractional and Cartesian coordinates

The relationship between fractional and Cartesian coordinates:

Similarly, the Cartesian coordinates:

Determination of cell properties in two and three dimensions using the metric tensor

is sometimes used for calculations involving the unit cell and is defined (in matrix form) as:

In two dimensions,

In three dimensions,

in the unit cell can be determined from the relation:

within the unit cell can be determined from the relation:

within the unit cell can determined from the relation:

can be determined from the relation:

Relationship between atomic distances, lattice parameters and fractional atomic coordinates

The following formula between two atoms from the lattice parameters and atomic coordinates:

between them according to the cosine formula:

References